I have heard that you have to spend at least six months in a year in
US to remain eligible for naturalization. Is that really true?
No, that is not true. There is no requirement regarding amount of time you have to spend in the US in a year. The only requirements are the continuous residence requirement and the physical presence requirement.
You are ...
The Apply for Citizenship | USCIS site seems to contain the needed information.
Step 2. Determine if you are eligible to become a U.S. citizen.
Going through the Naturalization Eligibility Worksheet contained in the pdf
gives in great detail, with possible rules of exceptions, how your times outside the US will be treated
and if you still qualify.
When can I re-apply for the citizenship, there is no date mentioned on the letter received.
You can reapply as soon as you meet the criteria. If you don't move again to another state or USCIS district, and if you don't spend so much time outside the US that you fail to meet the physical presence requirement, then you can reapply on the three-month ...